High School Mathematics UDL Instructional Unit-Lesson 5

Lesson 5: Objective

Grade Span: 9 - 10	Content Area: Math - Geometry
Lesson 5 of the Unit	Approximate Time Needed: 90 minutes or two 45 minute blocks

Objectives: Identify and quantify attributes of the problem that need to be measured. Determine a pattern. Generalize relationships. Determine the percent of increase/decrease. Determine the precision of measurement.

Essential Question(s): What are the relationships among the measurements of dimensions, area, and perimeter in problem solving situations? How does changing the sides of a square affect the area? How can we use variable expressions to reflect relationships?

Materials Needed: Large and small grid graph paper Worksheets Lesson Vocabulary: Area Centimeter Foot Inch Length Meter Ratio Unit of Measure Unit Rate Width Yard

Lesson 5: Introduction – 15 minutes

A. Activate Previous Knowledge (unit rate) Review the dance floor problem from the previous lesson with the whole group. Pose a new problem and discuss vocabulary with the class. Alex and Aldo planted several square apple orchards. The table below shows the number of trees and the size of the orchards. Review the information presented in each part of the table. Ask students to analyze the relationship between the side lengths and number of trees. No calculations needed. Keep it general (e.g., in orchard 1, the number of trees to length in feet is 2:1; in orchard 4, the number of trees to feet is 1:1). Orchard number Length of each side(feet) Number of apple trees 1st 2nd 8 feet 4 trees 3rd 12 feet 9 trees 4th 16 feet 16 trees Multiple means of representation: Present real life problems using drawings, models, and video representations of orchards of various sizes. Multiple means of expression: Allow students to present ideas for problem solving using computer models, demonstrations, visuals, etc. Record problem solving ideas in different formats: mathematics journals, computer, premade or original graphic organizers, etc. Multiple means of engagement: Students may choose the type of orchard when presenting problem. Allow students to work individually or in small groups based on learning style.

Additional Considerations for Emerging Readers and Emerging Communicators

During review, students should refer to their math journals or notes. Be sure students have graphic and/or tactile representations of relevant vocabulary (area, perimeter, length, width) as well as related materials/drawings/object representations from previous lessons. Provide examples through pictures, videos, or tactile representations of orchards of various sizes Point out how much space trees need for maximum growth and production. Ask guiding questions: "Do they have enough space?" "Is there room for more trees?" Students attempt to add more trees to an object representation or virtual representation of the orchard.

B. Establish Goals/Objectives for the Lesson Inform students that they will make decisions about units and scales that are appropriate for problem solving situations involving mathematics within mathematics or across disciplines or contexts and: Identify and quantify attributes of the problem that need to be measured. Determine a pattern. Generalize relationships. Percent of Increase/ Decrease. Determine the precision of measurement. Multiple means of representation: Along with posting lesson objectives in the classroom, provide individual copies for students. Multiple means of expression: Allow students to record lesson objectives in different formats: mathematics journals, computer, premade or original graphic organizers, etc. Multiple means of engagement: Brainstorm ideas of how and when these skills might be relevant to "me."

Additional Considerations for Emerging Readers and Emerging Communicators

Provide students with keys words paired with symbols/images/ tactile representations (i.e., units of measure, inch, foot, yard, centimeter, and meter). Provide key words in lesson objective paired with symbols/images/tactile representations to record into mathematics journals. Students may use an electronic picture writer to record the lesson objectives. Provide students with photographs, models, or tactile representations of examples of situations in which these concepts are used.

Lesson 5: Body – 30 minutes

Direct Instruction and/or Facilitation of the Lesson During this portion of the lesson, students will generalize relationships and determine the appropriate scale to express the relationship between two quantities. Review students' ideas on how to solve the orchard problem. Alex and Aldo planted several square apple orchards. The table below shows the number of trees and the size of the orchards. Orchard number Length of each side(feet) Area of Each Orchard (ft2) Number of apple trees 1st $x$ $?$ $y$ 2nd $8ft$ $64ft^2$ $4\ trees$ 3rd $12ft$ $144ft^2$ $9\ trees$ 4th $16ft$ $256ft^2$ $16\ trees$ 5th $x$ $?$ $y$ $n$ Students determine the area of each orchard (e.g., The 2nd orchard has an area of 64 ft2 because 8 x 8 = 64). Given the number of apple trees in each orchard, students determine the square footage needed for each tree using ratios and proportions (e.g., $\frac{area\ of\ orchard}{number\ of\ trees} = \frac{area}{1\ tree}$ or the unit rate area per tree). Using the ratio from orchard 2, students determine the unit rate (e.g., $\frac{64^2}{4\ trees} = \frac{?^2}{1\ tree}\ or\ 64ft^2 \div 4\ trees = 16ft^2$ needed for each tree) and confirm that measurement is true for each orchard (i.e., $144 ft^2 \div 9\ trees = 16ft^2 and 256ft^2 \div 16\ trees = 16ft^2$). Given the measurements in the length of each side column, students determine the rate of change in the length of each orchard (i.e., ___, 8, 12, 16 is a +4 pattern). Note: Students work in pairs to answer parts 1 - 5 of the problem. Using the rate of change +4, students determine the length of each side (x) for orchards 1 and 5, and fill in the column of the table. Note: Use whole group discussion for part 6. Students use that information to determine the area of the 1st and 5th orchards (i.e., the 1st orchard has an area of 16 ft2 because 4ft x 4ft = 16ft2 and the 5th orchard has an area of 400 ft2 because 20ft x 20ft = 400ft2). Students use the fact that each tree needs 16 ft2 to determine how many trees can be planted in the 1st and 5th orchards using $\frac{area\ of\ orchard}{area\ per\ tree} = number\ of\ trees$ (i.e., for the 1st orchard, $\frac{16ft^2}{16ft^2} = 1\ tree$ and for the the 5th orchard, $\frac{400ft^2}{16ft^2} = 25\ trees$). Students graph the rate of change in the length of each side and the consequent number of trees for each orchard (i.e., (x, y) where x = length of each side and y = the number of apple trees. Multiple means of representation: Allow students to refer to their brainstorming notes during discussion. When discussing unit rate, provide familiar examples (e.g., miles per hour). Provide students with a copy of the word problem and the table under #1. Have drawings and manipulatives available for students to use. Multiple means of expression: Allow students to solve the problem by using formulas and/or models and record information into the tables using various formats: computer, premade or original graphic organizer, etc. Allow students to use a reference of formulas. Multiple means of engagement: Ensure all students are actively involved in their partnerships. Use scenarios related to students' interests. For example, if a student is interested in animals instead of orchard trees, the scenario could involve the rate of grazing area per horse. Use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

Refer to the brainstorming ideas for solving the problem. Include picture and/or tactile representations as needed. Provide students with copies of the table as well as word/picture/tactile representations of the words orchard and apple trees. Knowing that the orchards are square, students should determine that the length and width are the same. Students determine the area of orchard #2 by using the formula length x length = area or 8ft x 8ft = 64 ft2 and/or students may also draw the orchard on grid paper to determine the area. Students can also be provided with a manipulative model or virtual template of orchard #2 so they can determine the area by counting the units. Since the numbers will be quite large, provide students with a means to skip count to determine area. Students can be given units grouped by 8 and a calculator set up to add 8 so each time students place a row of units into the template, they hit enter on the calculator to add 8. Students stop when the template is filled and indicates the final number for area from the calculator. If students are using a computer program, the program would be set up in the same way as lessons 1 and 2. See Example: Manipulative worksheets or PowerPoint Lesson 5, Slide 1. Review the concept of unit rate (area/tree) or the amount of space needed per tree. If orchard #2 is 64 ft2 and has 4 trees, how many square feet is needed for one tree? Students should set up the ratio as $\frac{area\ of\ orchard}{number\ of\ trees} or \frac{64^2}{64\ trees} = \frac{?^2}{1\ tree}$ Allow students to review strategies used in lesson 3 for using ratios and proportions to solve problems. Students should remember that the equation must remain balanced and that whatever was done to the top portion must be done to the bottom. Since the numbers decrease, students would use division. $\frac{64^2}{64\ trees} = \frac{?^2}{1\ tree} or \frac{64^2}{64\ trees} = \frac{\div4\ 16ft^2}{\div4\ 1\ tree}$ Students use manipulatives to determine how to divide the trees evenly to have a group of one tree (divide by 4). Students should divide the orchard by four as well. See Example: PowerPoint Lesson 5, Slide 2. Instruct students to determine the rate of change in the unit length of each orchard. Students determine the pattern (x, 8, 12, 16, x; pattern is +4). Students draw each orchard and lay them on top of each other to determine how the side lengths change or add/subtract the difference between unit lengths of the consecutive orchards to determine that each orchard changes by 4 ft. Students use that information to determine the unit length of orchard #1 and orchard #5. For example: $length\ of\ orchard \#1 + 4ft = length of orchard \#22$ $x+4ft = 8ft$ $x+4ft-4ft=8ft-4ft$ $x=4ft$ For example:$length\ of\ orchard \#4 + 4 ft = length\ of\ orchard \#5$ $16 ft + 4 ft = x$ $20 ft = x$ See Example: PowerPoint lesson 5, Slide 3. Students use the information of unit length to determine the area of orchards #1 and #5. Students use given formula for a square (length x length = area) and/or draw the orchards on grid paper and count the squares to determine the area. Students use manipulatives of the rate of change by placing a unit length of four, starting at the end of the length and width on orchard #2 to determine the unit length for orchard #1. Students draw or model the orchards on grid paper and count the squares to determine the area. See Example: PowerPoint Lesson 5, Slide 4. Students use the rate of change by adding it onto the length of orchard #4 to determine the length of orchard #6. See Example: PowerPoint Lesson 5, Slide 5. Now that students have determined the unit rate, they determine the number of trees per orchard for orchard #5. Using unit length determined in step 3, students use the formulas and ratios to determine the area of the orchard and the number of trees that can be planted in the orchard. Students can also use grid paper to draw orchard #5 (based on the dimensions of 20 ft x 20 ft) and by using a cut out of the unit rate, determine how many trees can be planted. See Example: Lesson 5 or students use virtual manipulatives as in PowerPoint Lesson 5 Slide 2. Tell students they will be graphing the relationship between the size of the garden and the number of trees that can be planted. Provide students with a coordinate grid with the x- and y- axes labeled. Students must use the information from their tables to create ordered pairs and complete the graph. The columns from which the ordered pairs are created are already labeled as x and y, but they can also be highlighted as needed. Students count over (the run) for x and up (the rise) for y, or students find the matching number for x and move the point up to the matching number for y. See Example: PowerPoint Lesson 5, Slide 6. Important Consideration: For some students, the difficulty/complexity can be reduced by using only the first quadrant of the coordinate grid.

Lesson 5: Practice – 30 minutes

Provide similar problems and additional practice questions based on students' responses. For example, Casey and Liz want to plant their own square apple orchard. They decide to increase the sides of Alex and Aldo's 3rd orchard by 25%. If they keep the same area per tree, how many trees can they plant in their square orchard? Multiple means of representation: Provide students with a copy of the word problem and the table. Have drawings and manipulatives available for students to use. Multiple means of expression: Allow students to solve the problem by using formulas and/or models and record information into the tables using various formats: computer, premade or original graphic organizer, etc. Allow students to use a reference of formulas. Multiple means of engagement: Ensure all students are actively involved in their partnerships. Use scenarios related to students' interests. For example, if a student is interested in animals instead of orchard trees, the scenario could involve a rate of grazing area per horse. Use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

Provide students with copies of the problem paired with picture and/or tactile representations. Ensure students have the table used during instruction to refer to for solving this problem. Students determine the area of Casey's and Liz's orchard by multiplying the area of orchard \#3 by 1.25. Students may also use a drawing of orchard \#3 on grid paper and divide it evenly in fourths (or quarters) to determine what 25% more would be and combine the quarter representation with orchard \#3. See Example: Manipulatives or PowerPoint Lesson 5, Slides 7 & 8. Once students have determined the area of Casey and Liz's orchard, they can determine how many trees can be planted based on the unit rate of $\frac{16ft^2}{1\ tree}$ Students should use the same strategies and supports as they used previously.

Lesson 5: Closure - 15 minutes

a. Revisit/Review Lesson and Objectives Remind students that they were to make decisions about units and scales that are appropriate for problem solving situations within mathematics or across disciplines or contexts and: Identify and quantify attributes of the problem that need to be measured. Determine a pattern. Generalize relationships. Percent of increase/ decrease. Determine the precision of measurement. Multiple means of representation: Along with posting lesson objectives in the classroom, students may refer to their individual copies. Multiple means of expression: Students can share what they have learned in different formats: writing, drawing, creative expression, etc. Multiple means of engagement: Share ideas of how and when these skills might be relevant to "me."

Additional Considerations for Emerging Readers and Emerging Communicators

When reviewing the expected outcomes, have students refer to the lesson objectives they recorded in their mathematics journals or their electronic picture versions. Students use the information recorded in their journals to refer back to the lesson objectives' key words paired with images. From that information, they share what they have learned based on each of the expectations. For example, students may grab the tactile cue for area to state "I have learned that the area is all the space measured within a figure." Students should also refer back to the photographs of examples of real-life situations when these concepts are used to share when they could use these new skills. For example, students could touch the tactile representations for area and orchard to state, "I can use the unit rate of area to tree to determine the size of the orchard."

B. Exit Assessment Students work either in pairs or individually to produce their own word problems similar to the ones presented in this lesson. Once the problems are written, students identify the unit rate of their problem (e.g., area per tree or area per person). If time permits, students can trade problems and solve them as a review for another in-class activity. Multiple means of representation: Ensure students have the previous word problems from this lesson and/or lesson 4 to review and model. Have previous drawings, models, and manipulatives available for students to use. Multiple means of expression: Allow students to create the problem using various formats: computer, premade or original graphic organizer, models, etc. Allow students to use a reference of formulas. Multiple means of engagement: Ensure all students are actively involved in creating their problems. Encourage students to use scenarios related to their interests. For example, if a student is interested in animals instead of orchard trees, the scenario could involve a rate of grazing area per horse. Use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

Students should have access to a variety of picture representations and models as they brainstorm ideas for their problems. Provide choices of interest to students in picture/tactile format. Students review the information in previous problems and choose the key words paired with images to use when creating their problems or provide students with a template of a word problem that they can complete with key words and unit rate concepts.

Lesson 5: Resources

UDL_HS_Math_Lesson_5_Resources.pdf

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